3D-patterned inverse-designed mid-infrared metaoptics

Modern imaging systems can be enhanced in efficiency, compactness, and application through the introduction of multilayer nanopatterned structures for manipulation of light based on its fundamental properties. High transmission multispectral imaging is elusive due to the commonplace use of filter arrays which discard most of the incident light. Further, given the challenges of miniaturizing optical systems, most cameras do not leverage the wealth of information in polarization and spatial degrees of freedom. Optical metamaterials can respond to these electromagnetic properties but have been explored primarily in single-layer geometries, limiting their performance and multifunctional capacity. Here we use advanced two-photon lithography to realize multilayer scattering structures that achieve highly nontrivial optical transformations intended to process light just before it reaches a focal plane array. Computationally optimized multispectral and polarimetric sorting devices are fabricated with submicron feature sizes and experimentally validated in the mid-infrared. A final structure shown in simulation redirects light based on its angular momentum. These devices demonstrate that with precise 3-dimensional nanopatterning, one can directly modify the scattering properties of a sensor array to create advanced imaging systems.


Two-Photon Polymerization (TPP) Accuracy
Fabrication via TPP is a flexible and powerful method, but also has known challenges in printing accuracy [1]. We observe shrinkage of the structure, which is dependent on the height of the layer from the substrate. Material printed on the bottom layer is not able to shrink from its printed size because it is physically adhered to the substrate. The topmost layer is roughly 90% of the desired lateral size and the bottom layer is close to the expected size. We also observe dilation of the smallest features in the design. Designs were compensated for this effect by pre-eroding features in the STL file before printing. Finally, the Nanoscribe had a mismatch between the feature size in each lateral direction. This is not a limitation of TPP, but likely the result of astigmatism in the optical alignment of our specific tool.

Laguerre Gaussian Modes for Angular Momentum Splitter
A spatially varying field can carry orbital angular momentum (OAM). Discrete values of OAM, l, can be found in the Laguerre-Gaussian orthonormal basis for solutions of the paraxial wave equation [2]. We used a simplified set with p = 0, such that each mode was defined at its waist (z = 0) with spatial profile in cylindrical coordinates: where w 0 is the waist radius of the beam. We chose w 0 = 8.5µm to ensure the mode was confined to the device. Transmission plots shown are geometrically normalized against the transmission of this beam through the device aperture with no device present. We can further assign a spin angular momentum of the mode by choosing the handedness of its circular polarization. The following pairs of OAM values l and spin values s were used in the optimization: (l, s) = (−2, 1), (−1, 1), (1, −1), (2, −1). These states were assigned to quadrants starting with the top right (blue) and moving counterclockwise (green, red, magenta).

12-Layer Stokes Polarimetry Device
The polarimetry device in the main text consists of six 3µm layers and struggles to achieve equal contrast for all four analyzer states with the circular polarization state lagging the others. We speculate this may be due to lack of degrees of freedom in the thickness of device. As a comparison, we optimize a thicker device consisting of twelve 3µm layers to see if the solution will display better contrast for all analyzer states. In Fig. S3 we show the comparison of the thicker device to the original. While the quadrant transmission per analyzer state is slightly reduced, the contrast metric is improved for the circular polarization state without sacrificing the other analyzer state contrasts.

Polarimetry Splitting Bounds
We can model the Stokes polarimetry device as a linear system that projects an input Jones state describing the x-and y-polarized electric field components onto several analyzer states. The Jones polarization is a 2-dimensional complex vector. The four analyzer states for our device are specifically chosen Jones vectors. In Fig. S6, analyzer states correspond to |v i ⟩, where N = 4 for the device in the paper. We assume the device outputs into four spatially distinct modes |w k ⟩, such that we take them to be orthogonal (⟨w i |w k ⟩ = δ ik ). Specifically, we model each output mode as a focused spot in a different quadrant of the focal plane and thus we assume the lack of spatial overlap implies orthogonality to a good approximation. The functionality of the device is described by an operatorQ where projection of an input state on each analyzer direction modulates the amplitude of an outgoing mode. We writê Without loss of generality, we assume α i is real. Any complex phase can be included in output mode |w i ⟩.

Maximum transmission into each analyzer state
Next, we assume for simplicity that all states have the same projection efficiency, such that α i = α. The transmission bound will differ from the following if each state does not split at the same projection efficiency. Consider an arbitrary state |a⟩ and it's orthogonal complement |ā⟩. The action ofQ on |a⟩ iŝ Taking the vector magnitude squared of the resulting state Since ⟨w i |w k ⟩ = δ ik , the double sum reduces to Following this pattern, we also have Due to energy conservation, we cannot have gained any magnitude through applyinĝ Q on the state so ⟨a|Q †Q |a⟩ ≤ 1 and ⟨ā|Q †Q |ā⟩ ≤ 1. Summing these together, we get Because the Jones vector space is 2-dimensional, |a⟩ and |ā⟩ form an orthonormal basis, so by definition (| ⟨a|v i ⟩ | 2 + | ⟨ā|v i ⟩ | 2 ) = 1. Thus, the sum simply becomes If we assume α is the largest it can be, then α 2 = 2 N . For N = 4 as is the case for the device in this manuscript, α 2 = 0.5. Thus, the maximum transmission we can achieve for each analyzer state into its output mode is 0.5.

Minimum overlap between analyzer states
Given a maximum transmission efficiency of 0.5 for each analyzer state, we can set a minimum overlap, β, for Jones vector analyzer states used in the splitter. While the choice is not unique, a maximally spaced set of vectors will have a common mutual overlap. Assume for our set of analyzer states, Sending in an analyzer state to the devicê Taking the magnitude like before and using the orthogonality of the |w i ⟩ states Using the common overlap between states in the analyzer set and requiring that by energy conservation this magnitude squared is bound by 1, The relation between α and β, then is given by Suppose we specialize to the case where the transmission is maximized into each analyzer state (α 2 = 2 N ) and we have no lost transmission for any given analyzer state through the system (⟨v k |Q †Q |v k ⟩ = 1). Then, Note the case of N = 2 requires no overlap between the vectors with β 2 = 0 and α 2 = 2 N = 1 because that matches the dimensionality of the Jones vector space. However, from two measurements, we cannot reconstruct the full Stokes vector where in order to do so we need at least N = 4. As stated before, for N = 4, α 2 = 0.5 at best and with no lost transmission for the analyzer states, β 2 = 1 3 .

Polarimetry Contrast Bounds
The contrast figure of merit for the Stokes polarimetry device is independent of overall transmission. For a given quadrant corresponding to analyzer state |v i ⟩ and orthogonal complement |v i ⟩, the contrast is related to the analyzer transmission T analyzer and orthogonal transmission T orthogonal to the quadrant as C = T analyzer −T orthogonal T analyzer +T orthogonal . In order to get a contrast of C = 1, we need to be able to completely extinguish light in the analyzer quadrant for the orthogonal state.

Analyzer state transmission to all quadrants
We first show that a given analyzer state must necessarily appear in more than just the desired quadrant. Following from the notation above, the action of the device on an analyzer state, |v k ⟩ is given bŷ We ask how much overlap does this have with one of the output modes |w j ⟩ not corresponding to the analyzer quadrant (i.e. i ̸ = j).
where we used ⟨w j |w i ⟩ = δ ij to eliminate the sum. However, as we showed above, with four analyzer states, ⟨v j |v k ⟩ ̸ = 0 even for j ̸ = i. So there is energy in the other quadrants according to the splitting efficiency of the j th analyzer state and the overlap between the j and k analyzer states.

Extinguishing orthogonal state to analyzer quadrant
We now check if an orthogonal state can be completely extinguished to the analyzer quadrant, which will determine if we can achieve a contrast of C = 1. When we send in the orthogonal state to a given analyzer, |v k ⟩, the device output is given bŷ Since it is true that ⟨v k |v k ⟩ = 0 by definition, the sum is reduced tô Now, we ask how much overlap does this have with the output mode corresponding to this analyzer quadrant, |w k ⟩, since we are interested in seeing if this overlap can be zero.
where ⟨w k |w i ⟩ = δ ki is only nonzero for i = k, but the sum explicitly ranges over values of i ̸ = k. Thus, we can extinguish a quadrant completely for a given orthogonal state and a contrast of 1 is theoretically achievable even if we transmit all incident light through the device to the focal plane.

Polarimetry Analyzer States
The choice of analyzer states that fits the above criteria is not unique, but will correspond to a tetrahedron with points lying on the Poincaré sphere. First, we choose evenly spaced pure polarization states in Stokes space and then evaluate their mutual overlaps in Jones space. One state is fixed in Stokes space to be right circular polarization (RCP), which is encoded as 1, 0, 0, 1 . This choice is arbitrary and different starting states will generate equally suitable sets of analyzer states. Staying on the Poincaré sphere surface means the first entry is fixed to 1 (from here, we write the vector in terms of S 1 , S 2 , and S 3 ). The other three states should lie on a circle with a fixed polar angle from this first state such that all mutual overlaps are the same. For polar angle θ and azimuthal angle ϕ, these states can be parameterized sin θ cos ϕ, sin θ sin ϕ, cos θ . To evenly spread out these states azimuthally, the spacing should be ∆ϕ = 2π 3 . We make the non-unique choice to set the first ϕ = 0. The first two states on the circle, then are sin θ, 0, cos θ and sin θ cos 2π 3 , sin θ sin 2π 3 , cos θ . Evaluating the dot product between any of the states on the circle and the right circular polarization state yields cos θ. The first two states on the circle have a dot product of sin 2 θ cos 2π 3 + cos 2 θ. Equating these two values generates the relation: Solving for cos θ gives cos θ = − 1 3 . Completing the tetrahedron, the final Stokes states (rounded to the thousands place) are: The squared overlap magnitudes between any of these states, β 2 = 1 3 as desired for equally split analyzer states.

Device Index of Refraction Profiles
Optimized index of refraction profiles for the multispectral and angular momentum sorting devices are shown in Fig. S7 and those for the Stokes polarimetry device from the main text and the one from the supplement with more layers are shown in Fig. S8.

Polarimetry Reconstruction
The following section shows how the polarimetry device presented in the main text can be used to recover the Stokes parameters of arbitrarily polarized inputs. This addresses interpretation of quadrant outputs when the excitation is different than the four analyzer states used in the design. It further addresses the ability of the device to utilize the four measurements to recover the degree of polarization for partially polarized light. This exploration is done in simulation, but the same calibration and reconstruction procedure can be used experimentally as well.

Reconstruction Method
The problem of converting the signal in each of the four quadrants into the incident polarization state can be phrased as follows: where M is the forward model that maps the Stokes vector, S, to the observed quadrant transmissions, T. We utilize the common definition of the Stokes parameters: where E x , E y , E 45 , and E −45 are projections onto horizontal, vertical, 45-degree, -45-degree linear polarizations, respectively and E R and E L are projections onto right-and left-circular polarizations, respectively. To calibrate the device, we input each of these individual polarization components and observe the transmission into each of the four quadrants. Then, we form: transmissions under excitation by the the S α state. We solve for M by taking the pseudo-inverse of σ and applying it on the right side, M = τσ † . Then, we form the solution or reconstruction matrix by taking the inverse of M, such that given a set of measurements T, we compute the Stokes parameters as S = M −1 T. We note this calibration could alternatively be done with the four analyzer states used in the design and we expect the results would be similar.

Reconstructing Pure Polarization States
The reconstruction method applied to pure polarization states is shown in Fig. S9 for different amounts of added noise in the transmission measurements to simulate different signal-to-noise ratios in the sensor detection. For p added noise, we add a normally distributed random variable with a mean of 0 and a standard deviation equal to p * T avg where T avg is the mean transmission across the four quadrant transmissions. As can be seen for increasing noise, the S 3 parameter is the most susceptible to a reduced signal-to-noise ratio. This is likely due to the circular polarization analyzer state exhibiting the lowest contrast and the S 3 Stokes parameter being a direct measure of the handedness of the circular polarization in the input.

Reconstructing Mixed Polarization States
The use of four projective measurements means information about partially polarized input states is contained in the quadrant transmissions. To test our ability to recover this property, we consider the situation where the polarization vector input into the device is randomly changing. We input a series of random polarization states into the device, and average the resulting quadrant transmission values for each quadrant. From these averaged transmission values, we reconstruct the Stokes vector in the same way as above. This reconstructed vector is compared to the averaged Stokes vectors for all the states input into the device. The degree of polarization of the light . Fig. S10 shows the results of reconstructing mixed polarization states. As the number of averaged states increases, the degree of polarization starts dropping. When noise is added per averaged state (using the same type of distribution as above), the squared error for the reconstruction is highest for the smaller number of averaged states. As this number of states increases, the fluctuating noise term starts averaging to zero thus decreasing the overall effect of noise on the reconstruction. Fig. S11 and Fig. S12 demonstrate the behavior of the angular momentum sorting device for different values of spin and OAM, respectively, than the design states. In an optical communication application, controlling the behavior of the device at these alternate points will depend on the amount noise present and mode distortion between communication links. However, in an advanced imaging context where information about the scene is inferred through the spatially resolved projection of the input onto different angular momentum states, the response of the device to other mode inputs needs to be at least characterized if not explicitly designed for the given application. As a note, the optimization technique used here was not directed to explicitly minimize or control the behavior of the device under these other excitations. By adding more simulations to each iteration to capture the effect of illuminating with these other modes, we can compute a gradient that either enables control over the quadrant these other modes couple to or extinguishes their transmission.

Illumination with Different Spin Values
In Fig. S11, we observe the device behaves similarly upon a flip in the handedness of the circular polarization for each angular momentum state. This can be seen through similar contrast and transmission profiles albeit at lower overall values. Thus, the optimization solution for the device relied primarily on the different OAM values for splitting and does not have strong polarization discriminating behavior.